Reflectionless measures

Speaker: 

Christian Remling

Institution: 

University of Oklahoma

Time: 

Thursday, November 13, 2008 - 2:00pm

Location: 

RH 306

Reflectionless measures are interesting objects because
they arise as limiting measures of spectral measures of arbitrary
Schr"odinger operators with some absolutely continuous spectrum.
In this talk, I'd like to review the definition and some background
material and then discuss more recent work, joint with Alexei Poltoratski,
on reflectionless measures.

Diffusion of wave packets in a Markov random potential

Speaker: 

Yang Kang

Institution: 

Michigan State University

Time: 

Thursday, November 6, 2008 - 2:00pm

Location: 

RH 306

In this talk, we consider the evolution of a tight binding wave packet propagating in a time dependent potential. We assume the potential evolves according to a stationary Markov process and show that the square amplitude of the wave packet converges to a solution of a heat equation. This is joint work with Jeff Schenker.

Eigenvalue Statistics for Random CMV Matrices

Speaker: 

Assistant Professor Mihai Stoiciu

Institution: 

Williams College

Time: 

Thursday, October 2, 2008 - 2:00pm

Location: 

RH 306

CMV matrices are the unitary analogues of one-dimensional discrete Schrodinger operators. We consider CMV matrices with random coefficients and we study the statistical distribution of their eigenvalues. For slowly decreasing random coefficients, we show that the eigenvalues are distributed according to a Poisson process. For rapidly decreasing coefficients, the the eigenvalues have rigid spacing (clock distribution). For a certain critical rate of decay we obtain the circular beta distribution.

Singular spectrum for Schrodinger operators generated by interval exchange transformations

Speaker: 

Associate Professor David Damanik

Institution: 

Rice University

Time: 

Thursday, September 25, 2008 - 2:00pm

Location: 

RH 306

We discuss joint work with Jon Chaika and Helge Krueger. The main result concerns explicit criteria for the absence of absolutely continuous spectrum for Schrodinger operators whose potentials are generated by an interval exchange transformation. In particular, we provide the first example of an invertible ergodic transformation of a compact metric space for which the associated Schrodinger operators have purely singular spectrum for every non-constant continuous sampling function.

On the Structure of Hofstadter's Butterfly

Speaker: 

Yoram Last

Institution: 

Hebrew University

Time: 

Thursday, August 28, 2008 - 2:00pm

Location: 

RH 440R

We review some aspects of the spectral theory of the critically coupled Almost Mathieu Operator connected with the structure of the famous associated "Hofstadter's Butterfly." We present a new result (joint with Mira Shamis) establishing that for a topologically generic set of irrational frequencies, the Hausdorff dimension of the spectrum of the critical Almost Mathieu Operator is zero. This result is based a new approach which combines certain inductive WKB-type estimates with Green function techniques and provides more detailed information than what has been previously achieved using more elaborate semiclassical approaches.

"Ghostbusting: Reviving quantum theories that were thought to be dead."

Speaker: 

Carl Bender

Institution: 

Washington University in St. Louis

Time: 

Thursday, May 1, 2008 - 2:00pm

Location: 

MSTB 254

The average quantum physicist on the street believes that a quantum-mechanical Hamiltonian must be Dirac Hermitian (symmetric under combined matrix transposition and complex conjugation) in order to be sure that the energy eigenvalues are real and that time evolution is unitary. However, the Hamiltonian $H=p^2+ix^3$, for example, which is clearly not Dirac Hermitian, has a real positive discrete spectrum and generates unitary time evolution, and thus it defines a perfectly acceptable quantum mechanics. Evidently, the axiom of Dirac Hermiticity is too restrictive. While the Hamiltonian $H=p^2+ix^3$ is not
Dirac Hermitian, it is PT symmetric; that is, it is symmetric under
combined space reflection P and time reversal T. In general, if a Hamiltonian $H$ is not Dirac Hermitian but exhibits an unbroken PT symmetry, there is a procedure for
determining the adjoint operation under which $H$ is Hermitian. (It is wrong to assume a priori that the adjoint operation that interchanges bra vectors and ket vectors in the Hilbert space of states is the Dirac adjoint. This would be like assuming a priori what the metric $g^{\mu\nu}$ in curved space is before solving
Einstein's equations.)

In the past a number of interesting quantum theories, such as the Lee model and the Pais-Uhlenbeck model, were abandoned because they were thought to have an incurable disease. The symptom of the disease was the appearance of ghost states
(states of negative norm). The cause of the disease was that the
Hamiltonians for these models were inappropriately treated as if they were DiracHermitian. The disease can be cured because the Hamiltonians for these models are PT symmetric, and one can calculate exactly and in closed form the appropriate adjoint operation under which each Hamiltonian is Hermitian. When
this is done, one can see immediately that there are no ghost states and that these models are fully acceptable quantum theories.

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